Mathematical Sciences: Presidential Young Investigator Award

这位总统青年研究员的数学研究的主要重点将是由几何问题引起的偏微分方程。方程是典型的非线性椭圆型和抛物线型。一类问题涉及到所谓的曲线缩短。问题是描述一组平面曲线的长时间行为，使它们的法向速度与曲率一致。当法向速度一般取决于曲率和时间时，会出现更复杂的情况；后一种情况发生在二维相变模型中。曲线缩短问题的解在某些情况下会产生奇点（膨胀）。（适当重新标度的）奇异部分的渐近形状不依赖于初始数据。数值模拟表明，曲率的实际爆炸速率是时间迭代对数的函数。将努力确定这一观察结果是否正确，并寻找导致估计长期存在的机制。在光滑紧致表面的共切束上确定具有规定能量的哈密顿系统的周期轨道方面也做了一些工作。假定哈密顿量在每条纤维上是凸的，并且是均匀的。该工作的一个特殊目标是建立能量表面上周期轨道的存在性，其在流形上的投影与给定周期轨道的投影相交规定次数。这样的结果可以看作是庞加莱-伯克霍夫定理的一个全局类比。相关工作将研究二维哈密顿系统部分解的联系。估计连接数可以看作类似于测量一个标量抛物型偏微分方程的根的个数。

The primary focus of the mathematical research of this Presidential Young Investigator will be on partial differential equations arising from geometrical problems. The equations are typically nonlinear of elliptic and parabolic type. One class of problems involves what is known as curve shortening. The question is one of describing the long time behavior of a family of plane curves which evolve so that their normal velocity coincides with their curvature. More complicated situations arise when the normal velocity depends in a general way on the curvature and time; the latter case occurring in models of phase transitions in two dimensions. The solution to a curve shortening problem has been shown to develop singularities (blow-up) under certain circumstances. The asymptotic shape of the (suitably rescaled) singular part does not depend on the initial data. Numerical simulations suggest that the actual blow-up rate of the curvature is a function of the iterated logarithm of time. Work will be done in determining if this observation is true and in seeking the mechanism responsible for the log-term in the estimate. Work will also be done in determining periodic orbits with prescribed energy of Hamiltonian systems on cotangent bundles of smooth compact surfaces. The Hamiltonian is assumed to be convex and even on each fibre. A particular goal of the work is to establish the existence of periodic orbits on energy surfaces whose projections on the manifold intersect the projection of a given periodic orbit a prescribed number of times. Such a result may be regarded as a global analog of the Poincare-Birkhoff theorem. Related work will study the linking of parts of solutions of Hamiltonian systems in two dimensions. Estimating linking numbers can be viewed as analogous to measuring the number of roots of a scalar parabolic partial differential equation.